When two chords of a circle bisect each other, then which of the following statements is true?
Both are diameters of the circle.
Step-1 Explanation for correct option:
(D) Both are diameters of the circle.
Diameters are the only chords of a circle that passes through the centre of a circle and they bisect every other chord inside the circle that they intersect with.
Hence when two chords of a circle bisect each other they are two diameters of the circle.
Step-2 : Explanation for incorrect options:
(A) When two chords of a circle bisect each other, then they need not be mutually perpendicular.
Let be the chords intersecting at P such that . As , we get . This means , i.e., the chords are of equal length. The perpendicular to and at their midpoints pass through the centre of the circle.
Unless is the centre of the circle this cannot happen. Thus and are diameters of the circle. They need not be mutually perpendicular.
(B) When two chords of a circle bisect each other, then Both chords cannot be parallel to each other.
(C) When two chords of a circle bisect each other, then they are two diameters of the circle Both chords cannot be unequal.
Consider the above diagram be the chords intersecting at P such that . As , we get . This means , i.e., the chords are of equal length.
Hence, option (D) is the correct answer.